Analytic origin of the Fisher–Huse autocorrelation exponents in the one-dimensional long-range Ising model

After a quench of the one-dimensional Ising model with power-law couplings $J(r)=r^{-(1+\sigma)}$ and nonconserved Glauber dynamics, the two-time spin autocorrelation obeys simple aging, $C(t,t_w)=f(L(t)/L(t_w))$, with only two universal scaling functions: for $\sigma>1$, where domain walls diffuse freely, $f(x)$ falls in the nearest-neighbour class with Fisher–Huse exponent $\lambda=1$; for $0<\sigma\le 1$, where wall motion is drift-dominated (biased), a new universal function appears with $\lambda=1/2$. These two values saturate the one-dimensional bounds $1/2\le\lambda\le 1$. Corberi, Lippiello and Politi (JSTAT 2019), who established this numerically, explicitly left open (i) whether the switch from symmetric to biased domain-wall diffusion is the essential and sufficient ingredient that pins $\lambda$ to its extremal values, or a coincidence, and (ii) the construction of a simple, physically oriented analytic derivation of the two exponents. Exact results exist for the long-range voter model but explicitly do not transfer to the Ising case, numerical confirmations and two-dimensional analogues abound, and the authors' own 2026 reviews still quote $\lambda=1/2$ as a purely numerical finding. The problem remains open.

Background

When a ferromagnet is quenched from a disordered phase to a temperature below its critical point (or to very low $T$ when $T_c=0$), it separates into ordered domains whose typical size $L(t)$ grows in time — the process of phase ordering, or coarsening. For the nearest-neighbour (nn) one-dimensional Ising model with single-spin-flip Glauber dynamics (a nonconserved order parameter), Roy Glauber's exact solution gives $L(t)\sim t^{1/2}$, and the two-time spin autocorrelation $C(t,t_w)=\langle s_i(t)s_i(t_w)\rangle$ exhibits simple aging: $C(t,t_w)=f(L(t)/L(t_w))$ with, at $T=0$, the exact scaling function $f_{nn}(x)=(2/\pi)\arcsin\sqrt{2/(1+x^2)}$ for $x\ge 1$ (Bray 1990; Prados, Brey and Sánchez-Rey 1997). For $x\gg 1$, $f(x)\simeq x^{-\lambda}$, which defines the Fisher–Huse exponent $\lambda$, introduced by Fisher and Huse (1988) through scaling arguments originally developed for spin glasses; the nn 1D value is $\lambda=1$. In any dimension these arguments bound the exponent, $d/2\le\lambda\le d$, the lower bound $\lambda\ge d/2$ having also been derived from the properties of the structure factor by Yeung, Rao and Desai (1996); in $d=1$ the allowed window is $1/2\le\lambda\le 1$.

Long-range couplings change the interfacial kinetics. For $J(r)=r^{-(1+\sigma)}$ with $\sigma>0$ (later literature often writes $J(r)\propto r^{-\alpha}$ with $\alpha=1+\sigma$), a domain wall (DW — the point-like boundary between a + and a − domain) feels a force from the other walls: in an overdamped picture the force on two walls a distance $L$ apart scales as $F\sim\sigma L^{-\sigma}$, so for $\sigma\le 1$ walls drift toward their nearest neighbour and a domain closes in a time $t\sim L^{1+\sigma}$. This drift-dominated (Rutenberg–Bray) regime has dynamical exponent $z=1+\sigma$ (defined by $L(t)\sim t^{1/z}$) and asymptotically asymmetric (biased) wall hopping; for $\sigma>1$ the drift is asymptotically slower than symmetric hopping and hence irrelevant, so walls hop symmetrically and diffuse freely, giving $z=2$ as in the nn model. At strictly $T=0$ the Glauber rate makes walls move deterministically toward each other at constant speed (ballistic regime, $z=1$).

Corberi, Lippiello and Politi (JSTAT 2019) simulated this model with large systems ($N=4\times 10^6$ spins, using interface-only Glauber dynamics in which only spins adjacent to a domain wall may flip) and found that $C(t,t_w)$ still obeys simple aging but with only two universal scaling functions. For $\sigma>1$, $f(x)$ is indistinguishable from the exact nn form ($\lambda=1$). For all $0<\sigma\le 1$, $f(x)$ is a new universal function, independent of $\sigma$ and of $T$ within the coarsening stage, with $\lambda=1/2$. The marginal case $\sigma=1$ is diagnostic: there $z=2$, exactly as in the diffusive regime, so $L(t)$ alone cannot distinguish the two kinetics, yet the scaling function is the $\sigma\le 1$ one with $\lambda=1/2$ — the autocorrelation detects the asymmetry of the wall motion. A quench to $T=0$ (ballistic DW motion), analysed but not shown in that paper, also gave $\lambda=1/2$. Since in $d=1$ the observed values coincide with the extremal allowed values $\lambda_{\min}=d/2=1/2$ and $\lambda_{\max}=d=1$, the authors conjectured that the switch from symmetric to biased DW diffusion is the ingredient that pins $\lambda$ to its bounds. They closed the paper by stating that they do not know whether this is a coincidence, and that such an interpretation, as well as a simple and physically oriented derivation of the Fisher–Huse exponents, would be very welcome. No such derivation exists: unlike the nn Glauber chain and, as discovered later, the long-range voter model, the long-range Ising model admits no closed exact equation for the autocorrelation (this is stated explicitly by Corberi and Smaldone in 2024), which is precisely why the exponents remain unexplained.

Problem Statement

Explain, analytically and physically, the Fisher–Huse autocorrelation exponents of the one-dimensional Ising model with power-law couplings $J(r)=r^{-(1+\sigma)}$, $\sigma>0$, and nonconserved Glauber dynamics: $\lambda=1$ for $\sigma>1$ and $\lambda=1/2$ for $0<\sigma\le 1$, independent of temperature within the coarsening stage, together with, ideally, the two associated universal scaling functions $f(x)$ of $C(t,t_w)=f(L(t)/L(t_w))$. Concretely:

(1) Provide a simple, physically oriented analytic derivation of both exponent values (and, ideally, of the full scaling functions) from the model's domain-wall kinetics.

(2) Determine whether the transition from symmetric (unbiased) to biased (drift-dominated) domain-wall diffusion is the essential and sufficient ingredient that pins $\lambda$ to its extremal one-dimensional values $\lambda_{\min}=d/2=1/2$ and $\lambda_{\max}=d=1$, or whether the observed saturation of the bounds is coincidental and additional mechanisms intervene.

The common scientific objective is a physical theory that predicts, rather than merely fits, the observed universal $\lambda$ values from the microscopic interfacial dynamics; items (1) and (2) are two linked faces of that objective and jointly define its completion.

References

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